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A transshipment problem (G , d , λ) is modeled by a directed graph G = (V , E) with weighted vertices (nodes) d = (d v ∣ v ∈ V) and weighted directed edges (arcs) λ = (λ e ∣ e ∈ E) interpreted as follows: G is a communication or transportation network, for example, a pipeline; each arc e ∈ E is a one-way communication line, road or pipe of capacity λ e , while every vertex v ∈ V is a node of production d v > 0 , consumption d v < 0 , or just transition d v = 0. A non-negative flow x = (x e ∣ e ∈ E) is called weakly feasible if for each v ∈ V the algebraic sum of flows, over all directed edges incident to v , equals d v , or shorter, if A G x = d , where A G is the vertex-edge incidence matrix of G. A weakly feasible flow x is called feasible if x e ≤ λ e for all e ∈ E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities x e > λ e are allowed. However, such an excess is viewed as unwanted (dangerous) and so we minimize the excess ratio vector r = (r e = x e / λ e ∣ e ∈ E) lexicographically. More precisely, first, we look for weakly feasible flows minimizing the maximum of r e over all e ∈ E ; among all such flows we look for those that minimize the second largest coordinate of r , etc. Clearly, in | E | such steps we obtain a unique lexmin vector r. We show that the corresponding balanced flow is also unique and represents the lexmin solution for problem (G , d , λ). Using the Gale–Hoffman inequalities, we construct it in polynomial time, provided vectors d and λ are rational. For symmetric digraphs the problem was solved by Gurvich and Gvishiani in 1984. Here we extend this result to arbitrary directed graphs. Furthermore, we simplify the algorithm and proofs applying the classic criterion of existence of a feasible flow for (G , d , λ) obtained by Gale and Hoffman in late 1950s. [ABSTRACT FROM AUTHOR]

Identifying internal waves in complex flow fields is a long‐standing problem in fluid dynamics, oceanography and atmospheric science, owing to the overlap of internal waves temporal and spatial scales with other flow regimes. Lagrangian filtering—that is, temporal filtering in a frame of reference moving with the flow—is one proposed methodology for performing this separation. Here we (a) describe an improved implementation of the Lagrangian filtering methodology and (b) introduce a new freely available, parallelized Python package that applies the method. We show that the package can be used to directly filter output from a variety of common ocean models including MITgcm, Regional Ocean Modeling System and MOM5 for both regional and global domains at high resolution. The Lagrangian filtering is shown to be a useful tool to both identify (and thereby quantify) internal waves, and to remove internal waves to isolate the non‐wave flow field. Plain Language Summary: Ocean flows are a superposition of many different flow phenomena including eddies, jets, currents, and waves. Quantifying a particular phenomenon therefore requires a method to identify and separate that phenomenon from others in the flow, in order to be able to assess the associated energy and energetic exchanges. Here we present a new implementation of a method to identify a particular phenomenon known as "internal waves"—hourly to daily oscillatory motions which propagate three‐dimensionally throughout the ocean, driving mixing and thereby supporting the global ocean circulation. The method involves a combination of a coordinate transformation and a high‐pass temporal filter. Here we describe a newly released, freely available, efficient and user‐friendly open‐source Python package that implements the method. Key Points: A new open source parallelized Python package to perform Lagrangian filtering is now availableThe package implements an improved method for filtering internal waves from high‐resolution numerical model outputThe package may be applied to directly filter output from a number of common ocean models including MITgcm, Regional Ocean Modeling System and MOM5 [ABSTRACT FROM AUTHOR]

An iterative dynamic initialization method is presented to produce balanced initial conditions for whole atmospheric global modeling. In this method, a global hydrostatic numerical model is iteratively nudged toward ground‐to‐space wind and temperature profiles at specific date and time. Ground‐to‐space atmospheric profiles are obtained by fitting spline curves to reanalyses below the lower mesosphere and empirical model results in the upper atmosphere. An optimal nudging coefficient is determined by examining if reasonable structure of mesospheric gravity wave (GW) momentum forcing and residual mean meridional circulations can be obtained from balanced initial conditions. Estimated mesospheric GW momentum forcing is found to exhibit a distinctive structure with larger (smaller) values in the lower and upper mesosphere (in the midmesosphere), when compared with parameterized climatological forcing. The iterative dynamic initialization allows for dynamical balance among the model's prognostic variables and reduces excitation of spurious GWs and noises at initial time. However, theoretical imbalances, measured by the ellipticity of the nonlinear balance equation, are not completely eliminated in balanced flows, and they are found in narrow tropospheric frontal regions and over localized areas associated with the large‐scale instability in the midlatitude middle atmosphere. These imbalances are discussed in the context of their potential relation to generation of planetary‐scale and inertia GWs around the middle atmospheric and tropospheric jets. Plain Language Summary: This study reports a method of initializing a whole‐atmosphere global model that covers from the ground to the lower thermosphere. Through this initialization, a whole atmospheric state balanced with the model dynamics is effectively generated. Free coefficient required for the initialization is chosen by estimating gravity wave momentum forcing and meridional mass circulations in the mesosphere and lower thermosphere. In the generated balanced flow, theoretical imbalances, measured by the ellipticity of the nonlinear balance equation, are found. These imbalances can be related to the spontaneous generation of gravity waves and balanced and unbalanced large‐scale instabilities in the middle atmosphere. The large‐scale instabilities can be attributed to generation of the planetary‐scale and inertia gravity waves in the middle atmosphere. Key Points: Iterative dynamic initialization for whole atmospheric global modeling is presentedGravity wave momentum forcing and mass circulations are estimated to constrain nudging coefficientDeviations from nonlinear balance may occur in association with large‐scale instabilities [ABSTRACT FROM AUTHOR]

In recent works, we developed a model of balanced gas flow, where the momentum equation possesses an additional mean field forcing term, which originates from the hard sphere interaction potential between the gas particles. We demonstrated that, in our model, a turbulent gas flow with a Kolmogorov kinetic energy spectrum develops from an otherwise laminar initial jet. In the current work, we investigate the possibility of a similar turbulent flow developing in a large-scale two-dimensional setting, where a strong external acceleration compresses the gas into a relatively thin slab along the third dimension. The main motivation behind the current work is the following. According to observations, horizontal turbulent motions in the Earth atmosphere manifest in a wide range of spatial scales, from hundreds of meters to thousands of kilometers. However, the air density rapidly decays with altitude, roughly by an order of magnitude each 15–20 km. This naturally raises the question as to whether or not there exists a dynamical mechanism which can produce large-scale turbulence within a purely two-dimensional gas flow. To our surprise, we discover that our model indeed produces turbulent flows and the corresponding Kolmogorov energy spectra in such a two-dimensional setting. [ABSTRACT FROM AUTHOR]